Avatar is a Sanskrit word for “incarnation,” and refers frequently in Hindu mythology to the physical incarnations of the gods (particularly Vishnu). In popular culture, an avatar is something that represents the controller in another form; for example, the appearance of Jake Sully as an “avatar” in Na’vi form on Pandora in the movie. Sully isn’t actually incarnated on Pandora. He’s in stasis elsewhere, controlling this avatar with his mind.
Examples appear all over the religious and mythic world, however. It could be said that the god Odin from Norse mythology was an avatar when he appeared as a wandering wizard or an old man on the road, although strictly in such stories he was merely in disguise. Some scholars may even argue that Jesus was an avatar of God, as by definition He was born in earthly form while still retaining his essence as God.
The Hindu god Vishnu appears in a number of avatars throughout history, including as Matsya the fish, Kurma the tortoise, and perhaps most importantly as the hero Krishna. He also appears several times as a beggar or washerman in the Ponnivala story, but here again these are better thought of as disguises than as avatars, because he is not born into these forms but merely assumes their appearance.